Economic growth and economic development 67

Introduction to Modern Economic Growth which verifies the alternative expression for the wage rate in (2.5) Returning to the analysis with the general production function, the per capita representation of the aggregate production function enables us to divide both sides of (2.11) by L to obtain the following simple difference equation for the evolution of the capital-labor ratio: (2.16) k (t + 1) = sf (k (t)) + (1 − δ) k (t) Since this difference equation is derived from (2.11), it also can be referred to as the equilibrium difference equation of the Solow model, in that it describes the equilibrium behavior of the key object of the model, the capital-labor ratio The other equilibrium quantities can be obtained from the capital-labor ratio k (t) At this point, we can also define a steady-state equilibrium for this model Definition 2.3 A steady-state equilibrium without technological progress and population growth is an equilibrium path in which k (t) = k∗ for all t In a steady-state equilibrium the capital-labor ratio remains constant Since there is no population growth, this implies that the level of the capital stock will also remain constant Mathematically, a “steady-state equilibrium” corresponds to a “stationary point” of the equilibrium difference equation (2.16) Most of the models we will analyze in this book will admit a steady-state equilibrium, and typically the economy will tend to this steady state equilibrium over time (but often never reach it in finite time) This is also the case for this simple model This can be seen by plotting the difference equation that governs the equilibrium behavior of this economy, (2.16), which is done in Figure 2.2 The thick curve represents (2.16) and the dashed line corresponds to the 45◦ line Their (positive) intersection gives the steady-state value of the capital-labor ratio k∗ , which satisfies δ f (k∗ ) = (2.17) ∗ k s Notice that in Figure 2.2 there is another intersection between (2.16) and the 450 line at k = This is because the figure assumes that f (0) = 0, thus there is no production without capital, and if there is no production, there is no savings, and the system remains at k = 0, making k = a steady-state equilibrium We will 53